The walk the pair, or group, if you include the dog, take in Lorna Crozier's "Crossing Willow Bridge," proves to be more than just a simple walk.
The mother, "seventy-six/this year," has recently become "unsteady/on her feet." The dog, a black lab, "All energy and muscle/and too much love," bumps the pair's legs and smacks the mother's legs with the stick. He makes the walk difficult.
Instead of a short, easy, leisurely walk, then, the walk becomes a trek, a journey, difficult and risky. The mother, being unsteady, could easily fall on her own. But the dog makes the possibility that much more likely.
The speaker also sees the dog as having "more to do/with time, how it runs ahead and keeps returning"--the dog, literally, and time, figuratively. Plus, the pair walks over the bridge, then back again.
The difficulty, the running of time, going back and forth over the bridge, all combine to make this walk more than a walk--they make it a journey.
This is a poem about a woman walking out into a muddy field with her elderly mother (76 years old). They are going to see a newborn baby llama. When she says "our walk becomes a journey" she is talking about two things, I believe. First, walking with an older woman is usually very slow, so it takes a long time, and may seem like a journey. This would be a metaphor. On a deeper level, they are probably bonding and having good mother/daughter conversations, so this is a type of journey. The mother's life has been a journey, and the daughter is sharing part of the journey on the walk.
In the lines you cite:
Suddenly this winter she's unsteady
on her feet. He runs to her
with a stick, strikes her legs
as if he's a monk and she
a stubborn student, seventy‑six
There is an example of personification - the dog striking her legs with the stick. Only a person would be able to do this. There is a simile: "as if he's a monk" comparing the dog to a monk. There is imagery: a monk striking a stubborn student for not paying attention in a Jesuit School, for example, is compared to the dog striking the mother's leg. It probably recalls images of being in a religious school. Comparing the mother to a "stubborn student" without using the words "like" or "as" is a metaphor.